Gov. Mark Dayton has said he would sign the bill. After the vote, he posted online, "Join us tomorrow as we make history on the Capitol steps."

The bill follows a failed attempt last year to define marriage as between one man and one woman with an amendment to the state constitution.

The state House approved the measure last Thursday.

The divisive issue drew crowds to Washington earlier this year when the Supreme Court heard arguments on a federal and state marriage law. The court heard challenges to the federal Defense of Marriage Act, which defines marriage as between one man and one woman, and to California's Proposition 8, a ban on same-sex marriage approved by voters.

The 11th state to legalize same-sex marriage, Delaware, did so last week.